Best Known (154, 164, s)-Nets in Base 8
(154, 164, large)-Net over F8 — Constructive and digital
Digital (154, 164, large)-net over F8, using
- 3 times m-reduction [i] based on digital (154, 167, large)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (21, 25, 2796200)-net over F8, using
- s-reduction based on digital (21, 25, 4194301)-net over F8, using
- net defined by OOA [i] based on linear OOA(825, 4194301, F8, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(825, 4194301, F8, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(825, 8388602, F8, 4) (dual of [8388602, 8388577, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(825, large, F8, 4) (dual of [large, large−25, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(825, large, F8, 4) (dual of [large, large−25, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(825, 8388602, F8, 4) (dual of [8388602, 8388577, 5]-code), using
- appending kth column [i] based on linear OOA(825, 4194301, F8, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(825, 4194301, F8, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
- s-reduction based on digital (21, 25, 4194301)-net over F8, using
- digital (35, 41, 2796200)-net over F8, using
- s-reduction based on digital (35, 41, 2796201)-net over F8, using
- net defined by OOA [i] based on linear OOA(841, 2796201, F8, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(841, large, F8, 6) (dual of [large, large−41, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(841, large, F8, 6) (dual of [large, large−41, 7]-code), using
- net defined by OOA [i] based on linear OOA(841, 2796201, F8, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- s-reduction based on digital (35, 41, 2796201)-net over F8, using
- digital (85, 98, 2796200)-net over F8, using
- net defined by OOA [i] based on linear OOA(898, 2796200, F8, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(898, 8388601, F8, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(898, 8388602, F8, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- trace code [i] based on linear OOA(6449, 4194301, F64, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6449, 8388602, F64, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(6449, 8388602, F64, 13) (dual of [8388602, 8388553, 14]-code), using
- trace code [i] based on linear OOA(6449, 4194301, F64, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(898, 8388602, F8, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(898, 8388601, F8, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(898, 2796200, F8, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- digital (0, 3, 9)-net over F8, using
- generalized (u, u+v)-construction [i] based on
(154, 164, large)-Net in Base 8 — Upper bound on s
There is no (154, 164, large)-net in base 8, because
- 8 times m-reduction [i] would yield (154, 156, large)-net in base 8, but